Final answer:
The answer considers the properties of isosceles trapezoids and congruent triangles to address geometric proofs for congruent segments and perpendicular lines. Without diagrams or further context, definitive proof cannot be given, but the relationships suggest potential congruence.
Step-by-step explanation:
The student's question pertains to various properties of geometry and vector mathematics. In the first part, congruent segments and perpendicular lines are crucial concepts. The first statement, where BC = EC and AC = DC, suggests an isosceles trapezoid, which might imply AB = DE by demonstrating that both are parallel and intercepted by the same perpendicularly dropped altitudes. However, without an accompanying diagram or additional information, a definitive proof cannot be provided.
The second statement involves a different scenario where AB and DC are both perpendicular to BC, thus forming two right-angled triangles. This implies that AB and DC are the legs opposite to the right angles, suggesting that triangles ABC and DBC might be congruent, which would in turn indicate that AB = DC.